The proportion of time an animal is in a feeding behavioral state.
Process Model
\[Y_{i,t+1} \sim Multivariate Normal(d_{i,t},σ)\]
\[d_{i,t}= Y_{i,t} + γ_{s_{i,g,t}}*T_{i,g,t}*( Y_{i,g,t}- Y_{i,g,t-1} )\]
\[ \begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix} \] \[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}} \] The behavior at time t of individual i on track g is a discrete draw. \[S_{i,g,t} \sim Cat(\phi_{traveling},\phi_{foraging})\]
Dive information is a mixture model based on behavior (S)
\(\text{Average dive depth}(\psi)\) \[ \psi \sim Normal(dive_{\mu_S},dive_{\tau_S})\]
## # A tibble: 11 x 3
## # Groups: Animal [?]
## Animal Year n
## <int> <ord> <int>
## 1 131111 2016 375
## 2 131115 2016 850
## 3 131116 2016 1475
## 4 131127 2016 6780
## 5 131128 2016 60
## 6 131130 2016 1014
## 7 131132 2016 1703
## 8 131133 2016 6419
## 9 131134 2016 1445
## 10 131136 2016 4874
## 11 154187 2016 1380
Dive profiles per indidivuals
Data Statistics before track cut
## # A tibble: 11 x 2
## Animal n
## <int> <int>
## 1 131111 58
## 2 131115 170
## 3 131116 294
## 4 131127 1327
## 5 131128 12
## 6 131130 70
## 7 131132 198
## 8 131133 1237
## 9 131134 279
## 10 131136 964
## 11 154187 275
## [1] 4884
Data statistics after track cut
## # A tibble: 10 x 2
## Animal n
## <int> <int>
## 1 131111 38
## 2 131115 170
## 3 131116 291
## 4 131127 1037
## 5 131130 59
## 6 131132 142
## 7 131133 1160
## 8 131134 240
## 9 131136 825
## 10 154187 269
sink(“Bayesian/ThreeState.jags”) cat(" model{
#from jonsen 2016
pi <- 3.141592653589
#for each if 6 argos class observation error
for(x in 1:6){
##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]
#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(firstmove[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
phi[i,g,t,1] <- Traveling[state[i,g,t-1]]
phi[i,g,t,2] <- Foraging[state[i,g,t-1]]
phi[i,g,t,3] <- 1-(phi[i,g,t,1] + phi[i,g,t,2])
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement in location
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
phi[i,g,steps[i,g],1] <- Traveling[state[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- Foraging[state[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],3] <- 1-(phi[i,g,steps[i,g],1] + phi[i,g,steps[i,g],2])
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])
#for each dive depth
#dive depth at time t
dive[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t]],depth_tau[state[i,g,t]])T(0.01,)
#dive speed (max depth/duration)
duration[i,g,t,u] ~ dnorm(duration_mu[state[i,g,t]],duration_tau[state[i,g,t]])T(0.01,)
#Assess Model Fit
#Fit dive discrepancy statistics
eval[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t]],depth_tau[state[i,g,t]])
E[i,g,t,u]<-pow((dive[i,g,t,u]-eval[i,g,t,u]),2)/(eval[i,g,t,u])
dive_new[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t]],depth_tau[state[i,g,t]])T(0.01,)
Enew[i,g,t,u]<-pow((dive_new[i,g,t,u]-eval[i,g,t,u]),2)/(eval[i,g,t,u])
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
tmp[3] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
theta[3] <- (tmp[3] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter)
##Behavioral States
gamma[1] ~ dbeta(4,2) ## 2d movement for traveling state
dev ~ dunif(0.1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2] <- gamma[1] * dev ## 2d movement for foraging state
gamma[3] ~ dbeta(2,20) ## 2d movement for resting state
#Temporal autocorrelation in behavior - remain in current state
Traveling[1] ~ dbeta(1,1)
Traveling[2] ~ dbeta(1,1)
Traveling[3] ~ dbeta(1,1)
#Temporal autocorrelation in behavior - transition to +1 state
Foraging[1] ~ dbeta(1,1)
Foraging[2] ~ dbeta(1,1)
Foraging[3] ~ dbeta(1,1)
#Probability of initial behavior
firstmove ~ ddirch(rep(1,3))
#Dive Mean Priors
depth_mu[1] ~ dnorm(0,0.0001)T(0,)
#we know that foraging dives are probably deeper than traveling dives,
forage ~ dunif(20,500)
depth_mu[2] <- depth_mu[1] + forage
#resting should basically be no dive depths (could be lnorm for exploratory dives)
depth_mu[3] ~ dnorm(0,0.01)T(0,)
#Duration Mean Priors
duration_mu[1] ~ dnorm(5,0.01) #traveling
time_forage~dunif(0,60)
duration_mu[2] = duration_mu[1] + time_forage #foraging
duration_mu[3] ~ dnorm(0,0.01)T(0,) # resting
#depth and duration variance
depth_tau[1] ~ dunif(0,500)
depth_tau[2] ~ dunif(0,500)
depth_tau[3] ~ dunif(0,500)
duration_tau[1] ~ dunif(0,500)
duration_tau[2] ~ dunif(0,500)
duration_tau[3] ~ dunif(0,500)
##Observation Model##
##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd
#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539
#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156
}"
,fill=TRUE)
sink()
## user system elapsed
## 3.136 0.290 220.523
Reconstructing the transition matrix
## user system elapsed
## 0.234 0.011 74.801
Where does the 3d predict foraging that the 2d misses?
Where does the 2d predict foraging that the 3d refines?
The goodness of fit is a measured as chi-squared. The expected value is compared to the observed value of the actual data. In addition, a replicate dataset is generated from the posterior predicted intensity. Better fitting models will have lower discrepancy values and be Better fitting models are smaller values and closer to the 1:1 line. A perfect model would be 0 discrepancy. This is unrealsitic given the stochasticity in the sampling processes. Rather, its better to focus on relative discrepancy. In addition, a model with 0 discrepancy would likely be seriously overfit and have little to no predictive power.
## # A tibble: 1 x 2
## `mean(E)` `var(Enew)`
## <dbl> <dbl>
## 1 -45.71297 27154632